Central limit theorem and a specific sequence

For a probability class I received the following exercise that I am supposed to solve with the help of the central limit theorem (CLT):

Let $X_1, \ldots, X_n$ and $Y_1, \ldots, Y_n$ both be identically distributed random variables with $\mathbb{E}[X_1] = \mu_1$, $\mathbb{V}[X_1] = \sigma_1^2 \in (0, \infty)$ and $\mathbb{E}[Y_1] = \mu_2$, $\mathbb{V}[Y_1] = \sigma_2^2$. Also let all random variables $X_1, Y_1, \ldots, X_n, Y_n$ be stochastically independent. Determine a sequence $(a_n)_{n=1}^{\infty}$ such that $$\lim\limits_{n \to \infty} P\big(\sum\limits_{j=1}^{n} X_j \leq a_n + \sum\limits_{j=1}^{n} Y_j\Big) = \frac{1}{2}$$

Although I know the central limit theorem, my attempts were not especially fruitful. What I tried first was to somehow rearrange the bracket insides and got

$$\lim\limits_{n \to \infty} P\big( \frac{\sum_{j=1}^{n} X_j - a_n}{\sum_{j=1}^{n} Y_n} \leq 1\Big) = \frac{1}{2}$$

which leads to nowhere since the CLT states that

$$\lim\limits_{n \to \infty} P\Big(\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \leq z\Big) = \Phi(z)$$

I have no idea how to connect this to the inequality given and would appreciate a hint.

• Try applying the CLT to $Z_i=X_i-Y_i$. – kimchi lover Nov 29 '17 at 13:38